Centers of maximal quotient rings
For a ring R, Z(R) and Q(R) denote the center and the maximal right quotient ring of R, respectively. In the paper the author considers the question: When is Z(Q(R)) the maximal right quotient ring of Z(R)? As the main result it is shown that a von Neumann regular ring R satisfies \(Z(Q(R))=Q(Z(R))\) if and only if R satisfies the following conditions: (1) Any non-zero ideal I of R which has no proper essential extensions in R contains a non-zero central element. (2) \(eZ(R)_{Z(R)}\) is essential in \(Z(eRe)_{Z(R)}\) for each idempotent e of R. Reviewer's comment: The author should make two points clear in stating (1): Do proper essential extensions mean in \(R_ R\), \({}_ RR\) or in both? Does I have to contain an non-zero central element of R or of itself?
scientific article; zbMATH DE number 4079650
scientific article